Active Spaghetti: Collective Organization in Cyanobacteria

Filamentous cyanobacteria can show fascinating examples of nonequilibrium self-organization, which, however, are not well understood from a physical perspective. We investigate the motility and collective organization of colonies of these simple multicellular lifeforms. As their area density increases, linear chains of cells gliding on a substrate show a transition from an isotropic distribution to bundles of filaments arranged in a reticulate pattern. Based on our experimental observations of individual behavior and pairwise interactions, we introduce a nonreciprocal model accounting for the filaments’ large aspect ratio, fluctuations in curvature, motility, and nematic interactions. This minimal model of active filaments recapitulates the observations, and rationalizes the appearance of a characteristic length scale in the system, based on the Péclet number of the cyanobacteria filaments.

Figure 1

A colony of O. lutea at density $\rho =53{\mathrm{mm}}^{-2}$ shows (a) a reticulate pattern, with (b) the local alignment of filaments within bundles, and (c) filament motion (arrows) that is predominantly parallel or antiparallel to neighbors.

Figure 2

Filament behavior. (a) The distribution of experimentally observed gliding speeds (blue) is well fit by a Gaussian (red, used for simulations). (b) Histogram showing how the alignment probability (left axis, blue bars) and interaction frequency (right axis, green circles) depend on incidence angle $\theta$. The data are experimental; see Supplemental Material [47], Sec. II—Interacting filaments, for more details. In the model, an incident chain is deflected on average by the relative angle $\mathrm{\Delta }\theta /\theta$ (black line). (c) In bundles, the directions of motion have a nematic distribution: a polar histogram compares experimental (blue) and simulated (red) cases. (d) Schematic of modeled interaction: when a chain’s head is within distance $d$ of another chain, it experiences an aligning effect.

Figure 3

Collective behavior and order-disorder transition. Panels (a)–(d) show micrographs of colonies at densities $\rho =25$, 31, 42, and $59{\mathrm{mm}}^{-2}$, respectively. Panels (e)–(h) show snapshots of simulations at comparable densities of $\rho =24$, 31, 41, and $59{\mathrm{mm}}^{-2}$. To avoid boundary effects, the simulated domains had sides $4.5\times$ larger than shown; panels are cropped to match the micrograph size. (i) Order parameter $⟨S⟩$, averaged over $1{\mathrm{mm}}^2$ blocks covering the experimental (blue) or simulated (red) domain; see Supplemental Material, Fig. S3 [47] for more details. Error bars and shading give the standard deviation of $⟨S⟩$ over the blocks. At low $\rho$ the filaments are randomly aligned, but locally nematic bundles and a reticulated structure emerge above $\rho \sim 40{\mathrm{mm}}^{-2}$.

Figure 4

Emergence of large-scale patterning. Finite-size scaling of the block-average order parameter $⟨S⟩$ was investigated in (a) simulations and (b) experiments, by varying the block size $l$ for the same data shown in Fig. 3. The power-law decay at low density indicates a disordered, isotropic state. The emergence of structures at high density is marked by a plateau lasting until $l$ reaches the size of the emerging structures, which we term the crossover length scale $l^{*}$, after which a more rapid decay is observed. (c) For different model parameters $l^{*}$ can be compared to the characteristic scale at which activity and fluctuations balance, ${\ell }^{*}$. Snapshots show the resulting patterns for some simulations with (d): $\rho =83{\mathrm{mm}}^{-2}$, $D_{\omega }=1.2\times {10}^{-9}{\mathrm{s}}^{-3}$, $\tau =480\mathrm{s}$. (e): $\rho =76{\mathrm{mm}}^{-2}$, $D_{\omega }=7.5\times {10}^{-10}{\mathrm{s}}^{-3}$, $\tau =1920\mathrm{s}$. (f): $\rho =83{\mathrm{mm}}^{-2}$, $D_{\omega }=7.5\times {10}^{-10}{\mathrm{s}}^{-3}$, $\tau =320\mathrm{s}$. (g): $\rho =69{\mathrm{mm}}^{-2}$, $D_{\omega }=1.7\times {10}^{-9}{\mathrm{s}}^{-3}$, $\tau =480\mathrm{s}$, and (h) for filaments growing naturally under typical incubation conditions. The characteristic scales of the patterns are shown by red circles of radius ${\ell }^{*}$. The scale bar in (d) also applies to (e)–(g).